Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 51-7;
http://www.math.u-szeged.hu/ejqtde/
A characterization of exponential stability
for periodic evolution families
in terms of lower semicontinuous functionals
1Constantin Bu¸
se
West University of Timi¸soara Department of Mathematics
Bd. V. Pˆarvan No. 4, 300223 Timi¸soara, Romˆania E-mail: buse@hilbert.math.uvt.ro
• 2000 Mathematics Subject Classification: 47A10, 47D03.
• Keywords and phrases: spectral radius of bounded operator, linear semigroup of operators, 1-periodic evolution family, exponential stabi-lity, lower semicontinuous functional.
Abstract
A characterization of exponential stability for a 1-periodic evolu-tion family of bounded linear operators acting on a Banach space in terms of lower semicontinuous functionals is given.
1
Introduction
LetT={T(t)}t≥0 be a strongly continuous semigroup on a Banach space X
andx∈X.We denote byfx the mapt7→ ||T(t)x||.The Datko-Pazy theorem is well known. It says that the semigroupTis uniformly exponentially stable if and only if for each x∈X the function fx belongs to Lp([0,∞)) for some
1 ≤p < ∞. See [3] for p = 2 and [9] for the general case. Jan van Neerven has shown that a part of this result still works replacing Lp([0,∞)) by any
Banach function space as in [7, Theorem 3.1.5].
Moreover an autonomous variant of Rolewicz theorem says that if φ : [0,∞) → [0,∞) is a nondecreasing function such that φ(t) > 0 for every
t >0 and if for each x∈ X the map φ◦fx belongs to L1([0,∞)), then T is
exponentially stable. Recently, Jan van Neerven obtained characterizations of exponential stability for semigroups in terms of lower semicontinuous func-tionals. The aim of this note is to extend the Neerven result from semigroups to periodic evolution families.
A family U = {U(t, s))}t≥s≥0 of bounded linear operators acting on a
Banach space X,is called a 1-periodic evolution family if: 1. U(t, t) =I (I is the identity operator on X); 2. U(t, s)U(s, r) =U(t, r) for every t ≥s ≥r≥0; 3. U(t+ 1, s+ 1) =U(t, s) for every t≥s≥0; 4. sup0≤s≤t≤1||U(t, s)||=M < ∞.
An 1-periodic evolution family U is called measurable if for each x ∈ X
the map t 7→ ||U(t,0)x|| is measurable. This notion will be used in the Corollary 2.3 below.
Practically one parameter semigroups will always occur with U(t, s) =
T(t−s).
It is easy to see that such family U has a growth bound, that is, there exist ω∈R and Mω ≥1 such that
||U(t, s)|| ≤Mωeω(t−s) for every t≥s≥0. (1)
We say that the family U is exponentially stable if we can choose a negative
ω such that the estimation (1) holds. Let V :=U(1,0) be the monodromy operator associated with the family U. It is well-known, see e.g. [1], that the family U is exponentially stable if and only if the spectral radius of the operator V satisfiesr(V)<1.
We denote by Mloc [0,∞) the space of all locally bounded functions on [0,∞) endowed with the topology of uniform convergence on bounded sets.
By M+loc [0,∞) we denote the positive cone of Mloc [0,∞). The fol-lowing result holds:
Theorem 1.1Let J :M+loc [0,∞)→[0,∞]be a map with the following properties:
1. J is lower semicontinuous;
2. J is nondecreasing, i.e. if f, g ∈ M+loc [0,∞) and 0 ≤ f ≤ g then
3. for each natural number k and any positive ρ there exists t ≥ 0 such that J(ρ·1[0,t])> k.
If a 1-periodic evolution family U is not exponentially stable then the set
{x∈X :J(||U(·,0)x||) = ∞}
is residual, that is, its complement is of the first category.
Here 1[0,t] is the characteristic function of the interval [0, t].
The proof of Theorem 1.1 is based on the following operator theoretical result which was proved by Jan van Neerven in [8]:
Theorem 1.2 Let T be a bounded linear operator on a Banach space X
and assume that its spectral radius satisfies r(T) ≥ 1. Then for all x ∈ X
and δ > 0 there exists a positive constant C with the following property: for all n ∈N (N is the set of all natural numbers) there exists y∈X such that
||x−y||< δ and ||Tjy|| ≥C for all j = 0,1,· · ·, n.
A natural consequence of Theorem 1.1 is the following result which ex-tends a similar one from [8]:
Theorem 1.3 Let J as in the above Theorem 1.1 and T = {T(t)}t≥0
be a locally bounded semigroup on a Banach space X, i. e. a semigroup of operators for which sup{||T(t)|| : t ∈ [0,1]} < ∞. If T is not exponentially stable, i.e. its uniform growth bound ω0(T) := inf
t>0
ln||T(t)||
t is nonnegative,
then the set
{x∈X :J(||T(·)x||) =∞}
is residual.
Particularly we can obtain the following generalization of a semigroup version of the Rolewicz theorem and of the Datko-Pazy theorem see for ex-ample [2], [10], [4], [7],[6], [5].
Theorem 1.4Let φ be a[0,∞)-valued, nondecreasing function on [0,∞)
such that φ(t) > 0 for all t > 0 and T = {T(t)}t≥0 be a locally bounded
set
{x∈X :
∞
Z
0
φ(||T(t)x||)dt=∞}
is residual.
Proof. The fact that T is locally bounded implies that for each x ∈ X
the map t 7→ ||T(t)x|| is measurable. Now we can apply the above Theorem 1.1 for the functional J(f) := ∞R
0 φ(f(t))dt defined for any locally bounded
and measurable function f on [0,∞).
2
Proof of Theorem 1.1 and other
consequences
Using Theorem 1.2 we can establish the following simple Lemma.
Lemma 2.1 If a1-periodic evolution family U is not exponentially stable (or equivalently if r(V)≥1) then for each x∈X and each positive constant
δ, there exists a positive constant Lwith the following property: for all t ≥0
there exists y∈X such that ||y−x||< δ and
||U(s,0)y|| ≥L for all s∈[0, t]. (2)
Proof of Lemma 2.1. Let C be as in Theorem 1.2, t ≥ 0, s ∈ [0, t] and
n = [s] + 1, where [s] denotes the integer part of s i.e. the biggest natural number which is less than or equal tos. We choosey as in Theorem 1.2 and we have:
C ≤ ||Vny||=||U(n,0)y||
= ||U(n, s)U(s,0)y||=||U(1, s−[s])U(s,0)y|| ≤ M||U(s,0)y||.
Proof of Theorem 1.1. We shall use the same ideas as in the proof of Theorem 4 in [8]. We may suppose that (1) is fulfilled with some positive ω.
Thus from (1), using the fact that each operatorU(t,0) is linear, follows the continuity of the map x 7→ ||U(·,0)x||. The lower semicontinuity of J and the continuity of the map x7→ ||U(·,0)x|| implies that for each k = 1,2,· · ·,
the set
Xk:={x∈X :J(||U(·,0)x||)> k}
is open. Then is suffices to prove that each Xk is dense in X.
Let x∈X and δ >0. We choose the constant Las in Lemma 2.1. Then for every k = 1,2,· · ·there exist tLk ≥0
and yk ∈X with ||yk−x||< δ and J(L·1[0,tLk])> k such that
||U(s,0)yk|| ≥L for every s∈[0, tLk].
Finally we obtain:
J(||U(·,0)yk||) ≥ J(||U(·,0)yk|| ·1[0,tLk])
≥ J(L·1[0,tLk])> k,
that is,yk ∈Xk.
In the end of this note we mention the following result which characterizes the exponential stability of periodic evolution families in terms of Banach function spaces.
Theorem 2.2 Let E be a Banach function space over [0,∞) such that
lim
t→∞||1[0,t]||E =∞
andU be a1-periodic evolution family which is not exponentially stable. Then the set
{x∈ X:||U(·,0)x||∈/ E}
is residual.
Corollary 2.3. Let E as in the Theorem2.2. We suppose that the norm ofEhas the Fatou property (see [8])and thatU is a1-periodic and measurable evolution family. If the set of allx∈X for which the map||U(·,0)x|| belongs to E is of second category in X, then U is exponentially stable.
Proof. The proof is modelled after [8]. Using the Fatou property follows that for each natural number k the set
Yk :={x∈X :|||U(·,0)x|||E ≤ k},
is closed. By assumption follows that ∪k≥1Yk is of the second category, so
there exists k0 such that Yk0 has nonempty interior. Moreover, if the open
ball with centre inxand radius 2δ >0 is contained inYk0 then by the triangle
inequality in E, the ball with centre in origin and radius 2δ is contained in
Y2k0. Then for each nonzero x∈X, we have:
|||U(·,0)x|||E =
||x||
δ |||U(·,0)( δ
||x||x)|||E <∞.
In order to obtain the assertion we apply Theorem 4.5 from [1].
ACKNOWLEDGEMENTS . The author thanks to Professor Constantin Niculescu for his comments on the preliminary version of this note. Also the author thanks the anonymous referees for their suggestions on preliminary version of this paper which have led to a substantial improvement in its readability.
References
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